We present the first -- of a series -- study of the evolution of galaxies in compact groups over the past 3 Gyr. This paper focuses on the evolution of the nuclear activity and how it has been affected by the dense environment of the groups. Our analysis is based on the largest multiwavelength compact group sample to-date, containing complete ultraviolet-to-infrared (UV-to-IR) photometry for 1,770 isolated groups (7,417 galaxies). We classified the nuclear activity of the galaxies based on optical emission line and mid-infrared diagnostic methods, as well as using spectral energy distribution fitting. We observe a 15% increase on the number of the AGN-hosting late-type galaxies found in dynamically old groups, over the past 3 Gyr, accompanied by the corresponding decrease of their circumnuclear star formation. Comparing our compact group results with those of local isolated field and interacting pair galaxies, we find no differences in the AGN at the same redshift range. Based on both optical and mid-IR colour classifications, we report the absence of Seyfert 1 nuclei and we attribute this to the low accretion rates, caused by the depletion of gas. We propose that the observed increase of LINER and Seyfert 2 nuclei (at low-zs), in the early-type galaxies of the dynamically young groups, is due to the morphological transformation of lenticular into elliptical galaxies. Finally, we show that at any given stellar mass, galaxies found in dynamically old groups are more likely to host an AGN. Our findings suggest that the depletion of gas, due to past star formation and tidal stripping, is the major mechanism driving the evolution of the nuclear activity in compact groups of galaxies.
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